Discrete Mathematics & Theoretical Computer Science, Vol 13, No 4

Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

Thomas P Hayes

Abstract

For every positive integer k, we construct an explicit
family of functions f: { 0, 1}n → { 0, 1} which
has (k+1)-party communication complexity
O(k) under every partition of the input bits into
k+1 parts of equal size, and k-party
communication complexity Ω(n / k4
2k) under every partition of the input bits into
k parts. This improves an earlier hierarchy theorem due
to V. Grolmusz. Our construction relies on known explicit
constructions for a famous open problem of K. Zarankiewicz, namely,
to find the maximum number of edges in a graph on n
vertices that does not contain Ks,t as a
subgraph.